picos.expressions.exp_sumxtr

Implements SumExtremes.

Classes

class picos.expressions.exp_sumxtr.SumExtremes(x, k, largest, eigenvalues=False)[source]

Bases: picos.expressions.expression.Expression

Sum of the k largest or smallest elements or eigenvalues.

Definition

Let k \in \mathbb{Z}_{\geq 1}.

  1. If x is an n-dimensional real vector or matrix and eigenvalues == False, then this is the sum of the k \leq n largest or smallest scalar elements of x, depending on the truth value of largest.

    Special cases:

    • If k = 1, this is either the largest element \max_{i = 1}^n \operatorname{vec}(x)_i or the smallest element \min_{i = 1}^n \operatorname{vec}(x)_i of x.

    • If k = n, this is the sum of all elements \langle x, 1 \rangle of x.

  2. If X is an n \times n hermitian matrix and eigenvalues == True, then this is the sum of the k \leq n largest or smallest eigenvalues of X, depending on the truth value of largest. Recall that the eigenvalues of a hermitian matrix are real.

    Special cases:

    • If k = 1, this is either the largest eigenvalue \lambda_{\max}(X) or the smallest eigenvalue \lambda_{\min}(X) of X.

    • If k = n, this equals the trace \operatorname{tr}(X).

If the given k exceeds the n of either case, then k is silently clipped to n.

__init__(x, k, largest, eigenvalues=False)[source]

Construct a SumExtremes.

Parameters

  • x (ComplexAffineExpression) – The affine expression to take a sum over.

  • k (int) – Number of summands.

  • largest (bool) – Whether to sum over the largest (eigen)values as opposed to the smallest.

  • eigenvalues (bool) – Whether to sum eigenvalues instead of elements.

property eigenvalues

Whether the sum concerns eigenvalues as opposed to elements.

property full

Whether the sum concerns all (eigen)values of the expression.

property k

Number of (eigen)values to sum.

property largest

Whether the sum concerns largest values as opposed to smallest.

property x

The expression under the sum.